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Approximation Algorithm for Vertex Cover with Multiple Covering Constraints Approximation Algorithm for Vertex Cover with Multiple Covering Constraints Eunpyeong Hong Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan [email protected] Mong-Jen Kao Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan [email protected] Abstract We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph G = (V, E) with a maximum edge size f, a cost function + w : V → Z , and edge subsets P1,P2,...,Pr of E along with covering requirements k1, k2, . , kr for each subset. The objective is to find a minimum cost subset S of V such that, for each edge subset Pi, at least ki edges of it are covered by S. This problem is a basic yet general form of classical vertex cover problem and a generalization of the edge-partitioned vertex cover problem considered by Bera et al. We present a primal-dual algorithm yielding an (f · Hr + Hr)-approximation for this problem, th where Hr is the r harmonic number. This improves over the previous ratio of (3cf log r), where c is a large constant used to ensure a low failure probability for Monte-Carlo randomized algorithms. Compared to previous result, our algorithm is deterministic and pure combinatorial, meaning that no Ellipsoid solver is required for this basic problem. Our result can be seen as a novel reinterpretation of a few classical tight results using the language of LP primal-duality. 2012 ACM Subject Classification Mathematics of computing → Approximation algorithms Keywords and phrases Vertex cover, multiple cover constraints, Approximation algorithm Digital Object Identifier 10.4230/LIPIcs.ISAAC.2018.43 Funding This work is supported in part by Ministry of Science and Technology (MOST), Taiwan, under Grants MOST107-2218-E-194-015-MY3 and MOST106-2221-E-001-006-MY3. 1 Introduction The vertex cover problem is one of the most well-known and fundamental problem in graph theory and approximation algorithms. Given an undirected hypergraph G = (V, E) and a cost function w : V → Z+, the objective is to find a minimum cost subset S ⊆ V such that any edge in E is incident to some vertex in S. This problem is known to be NP-hard, and f-approximation algorithms based on simple LP rounding and LP primal-duality are known for this problem [10], where f is the maximum size of the hyperedges. Assuming the unique game conjecture, approximating this problem to a ratio better than (f − ) is NP-hard for any > 0 [8]. The partial vertex cover problem is a natural generalization of the vertex cover problem. In this problem, we are given an additional parameter k which is called the covering requirement. The objective of this problem is to find a minimum cost subset of V which covers at least k edges in E, i.e., at least k edges of E are incident to at least one vertex in S. © Eunpyeong Hong and Mong-Jen Kao; licensed under Creative Commons License CC-BY 29th International Symposium on Algorithms and Computation (ISAAC 2018). Editors: Wen-Lian Hsu, Der-Tsai Lee, and Chung-Shou Liao; Article No. 43; pp. 43:1–43:11 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 43:2 On Vertex Cover with Multiple Covering Constraints Various methods have been developed to obtain tight approximations for this problem. Bshouty and Burroughs [3], who first proposed this problem, provided a 2-approximation algorithm for graphs, i.e., the case for which f = 2, using LP-rounding. Their algorithm generates |V | candidate covers each of which is constructed by guessing the most expensive vertex used in the optimal solution. Gandhi et a. [6] used the same technique to develop a primal-dual method which yields an f-approximation for hypergraphs. Mestre [9] used a more clever way to guess the most expensive vertex and improved the time complexity of the above algorithms. Fujito [5] developed an f-approximation for hypergraphs, based on a primal-dual method exploiting the property of minimal solutions. Bar-Yehuda [1] used the same property to obtain the same result using a local-ratio method. Hochbaum [7] adopted Lagrangian relaxation to get a 2-approximation on graphs. Bera et al. [2] considered a generalization of the partial vertex cover problem for which they called the partition vertex cover problem. In this problem, we are given a partition E1,E2,...,Er of the edges along with covering requirements k1, k2, . , kr. The objective is to find a minimum cost vertex subset that covers at least ki edges of Ei for each 1 ≤ i ≤ r. They obtained a (6c log r)-approximation for normal graphs, where c is a large constant used for Monte-Carlo randomized algorithms to ensure low error probability. They used randomized iterative rounding on a strong LP which is derived by knapsack inequalities on the natural LP. This approach generalizes to hypergraphs with an approximation guarantee of (3cf log r). They also showed that, even for normal graph for which f = 2, it is NP-hard to approximate this problem to a ratio better than Hr, which means O(f)-approximation for this problem is unlikely to exist. Wolsey [11] proposed the submodular set cover problem, which is a general formulation to the above covering problems, and presented an H(maxS∈S g({S}))-approximation, where g is the input submodular function and S is the ground set. Chuzhoy et al. [4] presented a simpler analysis to obtain a similar result. Fujito [5] presented a primal-dual algorithm for this problem which is useful for some special cases such as the partial vertex cover problem. Our Focus and Contributions In this paper, we consider the vertex cover problem with multiple covering constraints (VC-MCC) in hypergraphs. In this problem, we are given a hypergraph G = (V, E), a cost + function w : V → Z , and a number of covering constraints (P1, k1), (P2, k2),..., (Pr, kr), + where each Pi ⊆ E is a subset of E and ki ∈ Z is the covering requirement for Pi. The objective is to find a minimum cost subset S ⊆ V such that, for each 1 ≤ i ≤ r, at least ki edges of Pi are covered by S. This problem is a basic yet general form of classical vertex cover and a further generaliza- tion of the edge-partitioned vertex cover problem considered in [2]. In this paper, we present a primal-dual algorithm that yields an (f·Hr+Hr)-approximation for this problem, improving over the previous ratio of (cf log r) due to [2]. Our main contribution is the following theorem. I Theorem 1. There is a deterministic (f · Hr + Hr)-approximation algorithm for VC-MCC th which runs in polynomial time, where Hr is the r harmonic number. Compared to the previous result of (cf log r), our algorithm is deterministic and pure combinatorial, which means that our algorithm does not rely on heavy Ellipsoid LP solvers for this basic problem. Considering the lower-bound of Hr on the approximation ratio due to [2] and the well-known lower-bound of f for vertex cover, our result is much closer to the tight extent possible. E. Hong and M.-J. Kao 43:3 The novelty of this work lies in the way how we handle the dual variables. In contrast to previously known primal-dual approaches for covering problems, in which the dual variables can be handled freely, our approach manages the dual solutions carefully so that the following two criteria are met. 1. During the process, the cost of any vertex to be opened in the future must only be paid by the dual values possessed by the current unfulfilled covering constraints. 2. The overall dual value possessed by any unfulfilled covering constraint remains the same all the time. This makes the approximation guarantee of log r possible. Our result can be seen as a novel combination of the classical tight approximations with guarantees f and Hr for the covering problem, using the language of LP primal-duality. Our ingredient includes the strong LP relaxation due to [2], which is derived by applying Knapsack-cover inequalities to the natural LP. We would like to remark, however, that the usage of strong LP relaxation in our result is not a necessity but rather a better and more intuitive exposition of our ideas on how the dual variables can be managed, and obtaining the same result using natural LP is possible. Organization of this paper The rest of this paper is organized as follows. In Section 2, we define the notations we will be using throughout this paper and introduce the strong LP formulations. We present our approximation algorithm in Section 3 and conclude with future directions in Section 4. 2 Preliminary We use G = (V, E) to denote a hypergraph G with a vertex set V and an edge set E ⊆ 2V . Note that, under this notion, any edge e ∈ E is a subset of V that consists the incident vertices of the edge e. We use fG to denote the maximum cardinality of the edges in G, i.e., fG = maxe∈E |e|. The subscript G is omitted when no ambiguity is there in the context. For any edge subset M ⊆ E and any vertex v ∈ V , we use M(v) to denote the set of edges in M that are incident to v, i.e., M(v) := {e ∈ M : v ∈ e}. For any subset A ⊆ V , we use M(A) to denote the set of edges in M that are incident to the vertices in A, i.e., S M(A) = v∈A M(v).
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